Interference
Interference is the interaction of two or more waves
4.5.5 State the principle of superposition and explain what is meant by constructive interference and by destructive interference.
Principle of superposition: When two or more waves are superimposed, the net wave displacement is the vector sum of the individual wave displacements.
Constructive interference: Two waves with displacements that are in the same direction superimpose so that the resultant displacement is larger than either of the individual displacements of the two waves
Destructive interference: Two waves with opposite displacements superimpose and 'cancel' so that the resultant displacement is smaller than either of the individual displacements of the two waves
Click here to see an animation illustrating constructive and destructive interference.
More good animations here.
4.5.6 State and apply the conditions for constructive and for destructive interference in terms of path difference and phase difference.
Constructive interference:
- The displacements of interfering waves must be in the same direction
- Waves reaching a point are in phase (the crest of one wave coincides with the crest of the other, so that the resultant amplitude is the sum of the individual amplitudes)
Destructive interference:
- The displacements of interfering waves must be in opposite directions
- Waves reaching a point are out of phase (the crest of one wave will meet the trough of the other)
- If the waves are 180° out of phase and have the same amplitude, the crests and troughs of the two interfering waves will cancel exactly and form a node.
4.5.7 Apply the principle of superposition to determine the resultant of two waves
Simply add the displacements of the two waves at a point to get the resultant displacement
Standing Waves This webpage has a really good explanation of standing waves
11.1.1 Describe the nature of standing (stationary) waves Students should consider energy transfer, amplitude and phase
Standing waves are waves that appear to stay in the same position
Standing waves are created from two waves of equal amplitude, wavelength and frequency travelling in opposite directions
No net transfer of energy since the two waves carry equal energy in opposite directions
11.1.2 Explain the formation of one-dimensional standing waves Students should understand what is meant by nodes and antinodes
Two waves with the same frequency, wavelength and amplitude travelling in opposite directions will interfere and produce a standing wave.
For example, in a string attached to a wall at one end and being shaken on the other end, the waves generated by the shaking will hit the wall and be reflected back. If the reflected wave has the same amplitude, frequency and wavelength as the incoming wave, then the interference of the two waves will produce a standing wave.
Nodes are points which appear to stand still with no displacement. They are created by destructive interference.
Antinodes are like the opposite of nodes, moving between points of positive and negative amplitude and undergoing the greatest disturbance. Antinodes are created by constructive interference.
11.1.3 Discuss the modes of vibration of strings and air in open and in closed pipes. The lowest-frequency mode is known either as the fundamental or as the first harmonic. The term overtone will not be used.
This website explains frequency and harmonics very well and has many good images (where many of the images on this page come from and where Mr. Young appears to have taken his pics from)
Strings
The mode with the lowest possible frequency in a string is the first (or fundamental) harmonic. The modes increase in integers (i.e. the next mode after the fundamental harmonic would be the second harmonic, followed by the third etc.).
In the fundamental harmonic the length of the string is equal to half of the wavelength. In the following harmonics the length of the string increases by half a wavelength each time, so that for the second harmonic the length of the string would be two half (or one whole) wavelength and the length of the string of the third harmonic would equal 3 halves (or 1.5) wavelength
Each antinode represents half of a wavelength.
Since the speed stays constant, and the wavelength decreases in each successive harmonic, the frequency must increase.
The equation for the fundamental harmonic: f = v / 2L, where L is equal to half of the wavelength.
L is always equal to the number of the harmonic times 1/2, and the equation f = v / L is always multiplied by the same number (therefore for the fourth harmonic the equation would be f4 = 4v / 2L, where L = 4/2 wavelengths).
Because of harmonics the same note sounds different on different musical instruments,
2. Air Columns
All air columns are open at one end, but there are two different types of air columns: Open Pipe, where both sides are open, and a closed pipe, where only the one side in open.
In an open pipe there will be an antinode at both ends, while in a closed pipe, the wave-generating side (the closed side) will have a node while the open side will have an antinode.
The modes of vibration of an open pipe are the same as the ones for the string, however the standing wave picture looks different.
For closed pipes, the modes are different. The length of the pipe is only a quarter of the wavelength (wavelength = 4L). In each successive harmonic the wavelength increases by half. Therefore in the second harmonic L = 3/4 wavelength.
The harmonics of closed pipes increase in odd integers (1,3,5,7...).
This is interesting if you want to know more about harmonics of instruments - for all you music lovers!
and a video to go along with the musical instrument theme.
11.1.4 Compare standing waves and traveling waves.
A traveling wave travels through a medium, and produces a crest, which moves along from particle to particle. This crest is followed by a trough which is in turn followed by the next crest. A distinct wave pattern can be observed (in the form of a sine wave) traveling through the medium. This sine wave pattern continues to move in uninterrupted fashion until it encounters another wave along the medium or until it encounters a boundary with another medium. Traveling waves carry energy while standing waves do not.
A standing wave pattern is produced when one end of a string (thus the wave is confined to a given space) is vibrated. The interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium which appear to be standing still (antinodes and nodes).
Standing waves
Traveling waves
do not transfer energy
requires continual vibrating source
speed depends on medium
do transer energy
only require instant vibrating source
speed depends on medium
If you have more ideas for the table above, feel free to add.
11.1.5 Solve problems involving standing waves.
Use the equation f = v / L, and multiply the fraction by the number of the harmonic times half. L = (the harmonic * half) wavelength
Plug in variables you know to get the solution.
Remember that speed and L do not change, frequency increases and wavelength decreases with each harmonic
Don't forget to check if it is a string, open pipe or closed pipe.
for pratice questions go to Giancoli page 342 questions 1 to 10
The information below is for the interference of light shining through a double slit. The derivation and explanation are similar to single slit diffraction but keep in mind, they are not the same (single slit diffraction is in the core, double slit is in the optics option topic). The double slit experiment done by Thomas Young (no relation, well I am not sure) is one of THE experiments in physics. It showed conclusively that light behaved as a wave - temporarily winning the wave/particle argument (until Max Planck comes along). Thanks Aki (Mr DJY)
This isn't my page, but I found something cool so I'll post it. There was a lot of confusion about the derivation for the interference pattern of light and the angle at which the bands appear. I found a nice derivation by Walter Lewin, a physics professor from MIT. Here is the derivation that he did. I took it from a video, but it takes forever to load a 1 hour video so I typed it out for easy access :D
Walter Lewin's Derivation. This derivation is for a double slit. Rays from the bottom slit interfere with the rays from the top slit.
Interference
Interference is the interaction of two or more waves
4.5.5 State the principle of superposition and explain what is meant by constructive interference and by destructive interference.
- Principle of superposition: When two or more waves are superimposed, the net wave displacement is the vector sum of the individual wave displacements.
- Constructive interference: Two waves with displacements that are in the same direction superimpose so that the resultant displacement is larger than either of the individual displacements of the two waves
- Destructive interference: Two waves with opposite displacements superimpose and 'cancel' so that the resultant displacement is smaller than either of the individual displacements of the two waves
(Diagrams from http://www.sparknotes.com/testprep/books/sat2/physics/chapter17section4.rhtml)Click here to see an animation illustrating constructive and destructive interference.
More good animations here.
4.5.6 State and apply the conditions for constructive and for destructive interference in terms of path difference and phase difference.
- The displacements of interfering waves must be in the same direction
- Waves reaching a point are in phase (the crest of one wave coincides with the crest of the other, so that the resultant amplitude is the sum of the individual amplitudes)
- The displacements of interfering waves must be in opposite directions
- Waves reaching a point are out of phase (the crest of one wave will meet the trough of the other)
- If the waves are 180° out of phase and have the same amplitude, the crests and troughs of the two interfering waves will cancel exactly and form a node.
4.5.7 Apply the principle of superposition to determine the resultant of two waves
Standing Waves
This webpage has a really good explanation of standing waves
11.1.1 Describe the nature of standing (stationary) waves
Students should consider energy transfer, amplitude and phase
11.1.2 Explain the formation of one-dimensional standing waves
Students should understand what is meant by nodes and antinodes
For example, in a string attached to a wall at one end and being shaken on the other end, the waves generated by the shaking will hit the wall and be reflected back. If the reflected wave has the same amplitude, frequency and wavelength as the incoming wave, then the interference of the two waves will produce a standing wave.
11.1.3 Discuss the modes of vibration of strings and air in open and in closed pipes.
The lowest-frequency mode is known either as the fundamental or as the first harmonic. The term overtone will not be used.
This website explains frequency and harmonics very well and has many good images (where many of the images on this page come from and where Mr. Young appears to have taken his pics from)
2. Air Columns
This is interesting if you want to know more about harmonics of instruments - for all you music lovers!
and a video to go along with the musical instrument theme.
11.1.4 Compare standing waves and traveling waves.
A traveling wave travels through a medium, and produces a crest, which moves along from particle to particle. This crest is followed by a trough which is in turn followed by the next crest. A distinct wave pattern can be observed (in the form of a sine wave) traveling through the medium. This sine wave pattern continues to move in uninterrupted fashion until it encounters another wave along the medium or until it encounters a boundary with another medium. Traveling waves carry energy while standing waves do not.
A standing wave pattern is produced when one end of a string (thus the wave is confined to a given space) is vibrated. The interference of the incident wave and the reflected wave occur in such a manner that there are specific points along the medium which appear to be standing still (antinodes and nodes).
11.1.5 Solve problems involving standing waves.
- Use the equation f = v / L, and multiply the fraction by the number of the harmonic times half. L = (the harmonic * half) wavelength
- Plug in variables you know to get the solution.
- Remember that speed and L do not change, frequency increases and wavelength decreases with each harmonic
- Don't forget to check if it is a string, open pipe or closed pipe.
for pratice questions go to Giancoli page 342 questions 1 to 10Check out this link for harmonics in pipes etc. http://hyperphysics.phy-astr.gsu.edu/hbase/waves/opecol.html
The information below is for the interference of light shining through a double slit. The derivation and explanation are similar to single slit diffraction but keep in mind, they are not the same (single slit diffraction is in the core, double slit is in the optics option topic). The double slit experiment done by Thomas Young (no relation, well I am not sure) is one of THE experiments in physics. It showed conclusively that light behaved as a wave - temporarily winning the wave/particle argument (until Max Planck comes along). Thanks Aki (Mr DJY)
This isn't my page, but I found something cool so I'll post it. There was a lot of confusion about the derivation for the interference pattern of light and the angle at which the bands appear. I found a nice derivation by Walter Lewin, a physics professor from MIT. Here is the derivation that he did. I took it from a video, but it takes forever to load a 1 hour video so I typed it out for easy access :D
Walter Lewin's Derivation. This derivation is for a double slit. Rays from the bottom slit interfere with the rays from the top slit.
The actual video is here:
Simulations for Standing Waves:
Transverse (as in a string tied to a wall)
Longitudinal (as in the case of an instrument)
aki